H∞ model approximation for discrete-time Takagi–Sugeno fuzzy systems with Markovian jumping parameters

Author's Accepted Manuscript

H1 model approximation for discrete-time Takagi–Sugeno fuzzy systems with Markovian jumping parameters Xunyuan Yin, Xu Zhang, Lixian Zhang, Changhong Wang, Maryam Al-Yami, Tasawar Hayat

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PII: DOI: Reference:

S0925-2312(14)01739-1 http://dx.doi.org/10.1016/j.neucom.2014.12.075 NEUCOM15028

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Neurocomputing

Received date: 2 October 2014 Revised date: 4 December 2014 Accepted date: 26 December 2014 Cite this article as: Xunyuan Yin, Xu Zhang, Lixian Zhang, Changhong Wang, Maryam Al-Yami, Tasawar Hayat, H1 model approximation for discrete-time Takagi–Sugeno fuzzy systems with Markovian jumping parameters, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2014.12.075 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

H∞ Model Approximation for Discrete-Time Takagi–Sugeno Fuzzy Systems with Markovian jumping parameters Xunyuan Yin3 , Xu Zhang3 , Lixian Zhang1,2,3,∗ , Changhong Wang3 , Maryam Al-Yami2 and Tasawar Hayat2,4

Abstract This paper is concerned with the H∞ model approximation problem for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markov jump systems. The systems involve stochastic disturbances and nonlinearities that can be described by T-S fuzzy models. The problem to be solved in the paper is to find a reduced-order model, which is able to approximate the original T-S fuzzy Markov jump system with comparatively small and acceptable errors. Specifically, the corresponding error system is guaranteed to be asymptotically stable in the mean square with a prescribed H∞ performance index. By using convex optimization approach and projection approach, respectively, sufficient conditions on the existence for such model with reduced-order are obtained and presented in the form of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the obtained results. Keywords: H∞ performance, Markov jump systems, projection approach, Takagi–Sugeno (T–S) fuzzy systems, model order approximation, disturbances

1

State Key Laboratory of Urban Water Resources and Environment, Harbin Institute of Technology, Harbin 150090, China

2

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King

Abdulaziz University, Jeddah 21589, Saudi Arabia 3

School of Astronautics, Harbin Institute of Technology, Harbin, 150080, China

4

Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan

∗

Correpsonding author. e-mail: [email protected].

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I. INTRODUCTION It is constantly encountered that many practical engineering systems, regardless of the fields of their applications, are characterized by mathematical models with relatively high order, which will inevitably introduce difficulty and computational complexity to stability analysis, synthesis and simulation [1], [2]. Therefore, an effective solution to the underlying problem is to reduce the order of the system models to an comparatively low order. Thus, model approximation for engineering systems is playing a significant role in the process of control system analysis and design. Given a full-order engineering system, the objective in terms of model approximation is to achieve a new model with reduced order that is able to approximate the original system such that the errors are small and acceptable. In order to achieve these goals, effective approaches have been proposed, such as the aggregation method [3], the optimal Hankel norm approximation method [4], the balanced truncation method [5], etc. During the past several decades, much attention has been paid on model reduction for different kinds of systems, such as switched hybird systems [2], [6], Markov systems [7] and fuzzy systems [8]. On the other hand, great importance has been attached to the areas in terms of practical engineering systems with switching dynamics during the past several decades. These systems may experience inevitable changes or disturbances, such as unpredictable environmental changes, random component failures or other influences brought by interactions of subsystems [9]. The aforementioned systems that involve stochastic mode transitions can be represented by Markov jump systems, which have been considered to be a frequently-addressed topic due to the widely practical applications in terms of power engineering systems, communication systems, irrigation systems, aerospace engineering, networked control systems and other manufacturing systems [7], [10]–[13]. Involving stochastic processes, Markov jump systems are hybrid in essence, in which the continuous and discrete dynamics are, respectively, described by a series of differential equations and Markov chains to govern the transitions among each subsystem [14]. The phenomenon is known as a stochastic behaviour, which makes Markov systems vastly different from other kinds of hybrid systems, such as the nondeterministic switched systems. Markov jump system has attracted much attention mainly because of its promising prospect in many application areas. For instance, it can be adopted to describe the abrupt phenomenon such as random failures, repairs of the components and sudden environmental changes [13]. So far, many

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significant results on the stability analysis, filter design, optimal control, robust control, model reduction and other issues for Markov jump linear systems (MJLSs) have been obtained, such as [7], [15]–[18], and the references therein. Compared to the literatures on MJLSs, fewer papers can be found on control issues for Markov jump nonlinear systems (MJNSs) due to the complexity generated from the nonlinear dynamics. However, the problems in terms of MJNSs are of vital importance and should be combated successfully because nonlinear systems are ubiquitously found in chemical process, power engineering systems, robotic systems, automotive systems, and many other manufacturing systems [8]. Problems on approximating nonlinear systems with acceptable errors were deeply investigated. Due to the nature of high efficiency, the T-S fuzzy system has become very important for the analysis and synthesis of complex nonlinear systems [19]. The work on stability analysis and controller design for T-S fuzzy systems with complex structures, time delay, disturbances, etc. So far, T-S fuzzy model has presented excellent performance in facilitating stability analysis and controller design [20], [21], thus has attracted great attention. Some typical results on the stability analysis and controllability of T-S fuzzy systems are reported in [19], [21]–[24] and [25]–[27]. Because of this advantage of the T-S fuzzy model, the concept of fuzzy Markov jump systems (FMJSs) was proposed and has attracted much attention during the past few years. By utilizing a family of IF-THEN rules that represent local linear input-output relations of each model of the MJNSs, the problem on the nonlinear dynamics can be efficiently addressed. To name a few, a robust H∞ output feedback controller was designed in [28]. A mode-based method for FMJSs which can be found in [29] was proposed and verified. Stability analysis and stabilization issues of discrete-time FMJSs with time delays were investigated in [30] and [31]. A mode-independent fuzzy controller design method was reported in [32]. The stability analysis and a novel controller design technique for FMJSs was studied in [23], where slack variables were introduced to separate Lyapunov matrices from system matrices. Note that though the studies on control of FJMSs have been launched, the model approximation issue has seldom been addressed, which motivates us for this study. In this paper, the problem of H∞ model approximation for a class of discrete-time FMJSs is studied. Specifically, both convex linearization approach and projection approach are introduced to derive sufficient conditions of the existence of the reduced-order model and the corresponding solutions to system matrices of the reduced-order model such that the resulting error system is February 6, 2015

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said to be stochastically stable with a guaranteed H∞ performance index. Since the results based on the projection approach are presented with the form of LMIs with a non-convex constraint, cone complementary linearization (CCL) is employed to cope with the sequential minimization issue. Finally, a numerical example is presented to illustrate the validity and effectiveness of the proposed model reduction approach. The remained part of the paper is organized as follows. In Section II the formulation of the problems and preliminaries are presented. Main results on H∞ model reduction of FMJSs are presented in Section III, while simulation results are obtained in Section IV to demonstrate the effectiveness of the main results. Finally, the conclusion is drawn in Section V. Notation: The notation used throughout this paper is fairly standard. The subscripts “T ” and “−1” stand for matrix transposition and inverse, respectively. Rn denotes the n-dimensional Euclidean space. The notation P > 0 means that P is real symmetric positive definite. I and 0 represent identity matrix and zero matrix, respectively.. In symmetric block matrices or complex matrix expressions, the symbol “∗” is used to represent an ellipsis for the terms that are introduced by symmetry and diag{...} stands for a block-diagonal matrix. k·k denotes the Euclidean norm of a vector and its induced norm of a matrix. k·k2 stands for the typical l2 [0, ∞) norm while k·k∞ represents the l2 -induced norm of a transfer function matrix or a general operator. For a matrix U ∈ Rm×n with rank k, we denote U ⊥ as the orthogonal complement, which is possibly non-unique, such that U ⊥ U = 0. U + is defined as the Moore-Penrose inverse of the U , and UL , UR are defined as any full rank factors of U , i.e. UL UR = U . II. PROBLEM FORMULATION AND PRELIMINARIES A. Physical Plant Consider the following discrete-time FMJSs. Plant rule i: IF s1 (k) is µi1 and s2 (k) is µi2 and ... and sh (k) is µih , THEN   x(k + 1) = A (r )x(k) + B (r )u(k) i k i k  y(k) = C (r )x(k) + D (r )u(k) i

k

i

(1)

k

where x(k) ∈ Rn is the state vector, u(k) ∈ Rp which belongs to l2 [0, ∞) is the input and y(k) ∈Rq is the measured output; i ∈ S = {1, 2, ..., w} , w is the number of IF-THEN rules, µi1 , µi2 , ..., µih is the fuzzy set, s1 (k), s2 (k),..., sh (k) are the premise variables; {rk } is a discrete-time February 6, 2015

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Markov process that takes values in a finite set T = {1, 2, ..., g} with mode transition probability matrix Ψ = {π lm } that is listed as follows: π lm = Pr(rk+1 = m|rk = l) where π lm represents the transition probability from mode l at time k to mode m at time k + 1, P for all l ∈ T, m ∈ T, π lm ≥ 0 and gm=1 π lm = 1. The system matrices of different modes are denoted by Ai (rk ), Bi (rk ), Ci (rk ) and Di (rk ), which are real known matrices with appropriate dimensions. For notational convenience, in the subsequent sections of this paper, for r(k) = l, l ∈ T, we will denote the systems matrices that are related to the lth mode by: Ail = Ai (rk ), Bil = Bi (rk ), Cil = Ci (rk ), Dil = Di (rk ). It is assumed that the premise variables are not dependent on the input variables u(k). Then if given a pair of (x(k), u(k)), for any r(k) = l, l ∈ T, the final model of the fuzzy Markov jump system in (1) can be easily obtained as follows:  w X     x(k + 1) = hi (s(k)) {Ail x(k) + B il u(k)} i=1

    where

y(k) =

w X

(2)

hi (s(k)) {Cil x(k) + Dil u(k)}

i=1

h Y vi (s((k)) hi (s(k)) = w , vi (s((k)) = µif (sf (k)) X f =1 vi (s((k)) i=1

In (2), hi (s(k)) is the fuzzy basis function, µif (sf (k)) is the grade membership of sf (k) in P µif . Suppose vi (s((k)) ≥ 0 and w i=1 vi (s((k)) > 0 for all k. Therefore, we have hi (s(k)) ≥ 0 Pw and i=1 hi (s(k)) = 1 for all k. In this paper, we are interested in finding an effective method by which the original system (2) can be accurately approximated by a reduced-order mathematical model that is of the following structure:

 w n o X  ˆ ˆ  x ˆ (k + 1) = h (s(k)) A x ˆ (k) + B u(k)  i il il  i=1 w n o X   ˆ ˆ  y ˆ (k) = h (s(k)) C x ˆ (k) + D u(k) i il  il

(3)

i=1

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ˆil , Cˆil and D ˆ il , where xˆ(k) ∈ Rt is the state vector of the reduced-order system with t < n, Aˆil , B for all l ∈ T, are matrices with appropriate dimensions that are of compatible manipulations. These matrices in the reduced-order model are to be determined later. 4

4

By denoting x˜(k) = [ xT (k) xˆT (k) ]T , e(k) = y(k) − yˆ(k), we are able to augment the model of (2) to include (3), the error system is obtained as follows:  n o w P ˜ ˜  x ˜ (k + 1) = h (s(k)) A x ˜ (k) + B u(k)  i il il i=1 n o w P  ˜ u(k)  e(k) = hi (s(k)) C˜il x˜(k) + D il

(4)

i=1

where 

C˜il





0 B ˜il =  il  ,B ˆil B 0 Aˆil h i ˜ il = Dil − D ˆ il = Cil −Cˆil , D

A˜il = 

Ail



In order to present the main results of the paper in the sequel clearly and precisely, the following two definitions are required. More details on both of them can be found in [14] and the references therein. Definition 1: The fuzzy Markov jump system in (4) is said to be stochastically stable with u(k) ≡ 0, if for any initial condition x0 ∈ Rn and s(0) ∈ T, the following inequality holds: ( N ) X lim E xT (k, s(0))x(k, s(0)) < ∞ N →∞

k=1

Definition 2: Given a scalar γ > 0, the fuzzy Markov jump system in (4) is said to be stochastically stable and has an H∞ model error performance index γ if the system is stochastically stable and with all nonzero uk ∈ l2 [0, ∞), the following inequality is satisfied: (∞ ) ∞ X X T 2 E e (k)e(k) < γ uT (k)u(k) k=1

(5)

k=1

Now, we are in a position to describe the model approximation problem that will be addressed in this paper as follows: H∞ model approximation for FMJSs: Consider the model error system in (4). Given a performance index scalar γ > 0, design a new system model with reduced order to precisely describe the original system in the form of (3), such that

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1) (stochastic stability) the model error system (4) is stochastically stable in the mean square in the sense of Definition 1; 2) (H∞ performance) with zero initial condition, the system error e(k) satisfies kek2 ≤ γ kuk2 for all nonzero u ∈ l2 [0, ∞). If the aforementioned two conditions are both satisfied, we say that the T-S fuzzy system with Markov jumping parameters is stochastically stable with a guaranteed H∞ performance index. Before ending this section, it is necessary to introduce following useful lemmas which are indispensable for the proof process of the main results of the paper. Lemma 1: [33] For any real matrices X, Y, the following inequality holds: ∀β > 0 X T Y + Y T X ≤ βX T X + β −1 Y T Y Lemma 2: For any positive integer c, and Matrices Qr , Qv , where r, v ∈ [1, 2, .., c]. Further, P we define coefficients 0 < dr < 1 and cr=1 dr = 1, then the following inequality holds c c c X X X dr QTr Qr (6) dv Qv ≤ dr QTr · r=1

v=1

r=1

Proof. Firstly, we consider the LHS of (6). c c X X T dv Qv dr Qr · r=1

=

v=1 2 T d1 Q1 Q1 +

d1 d2 QT1 Q2 + d1 d2 QT2 Q1 ... + d1 dc QT1 Qc + d1 dc QTc Q1 + ... + d2c QTc Qc

By using Lemma 2, and let β = 1, we have d21 QT1 Q1 + d1 d2 QT1 Q2 + d1 d2 QT2 Q1 ... + d1 dc QT1 Qc + d1 dc QTc Q1 + ... + d2c QTc Qc ≤ d21 QT1 Q1 + d1 d2 QT1 Q1 + d1 d2 QT2 Q2 + ... + d1 dc QT1 Q1 + d1 dc QTc Qc + ... + d2c QTc Qc c c c X X X T T = d1 dr Q1 Q1 + d2 dr Q2 Q2 + ... + dc dr QTc Qc =

r=1 c X

r=1

r=1

dr QTr Qr

r=1

This ends the proof. Lemma 3: [7] Take consideration of system (4) with known transition probabilities. Then the system (4) is said to be stochastically stable if u(k) = 0, k ≥ 0, and there exist matrices Pl > 0, l ∈ T, such that

 

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−1

−λ

λAil

∗

−Pl

 <0

(7)

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where λl =

P

m∈T

π lm Pm .

Proof. Construct a Lyapunov function as follows: V (˜ x(k), r(k)) = x˜T (k)P (r(k))˜ x(k) where Pl , ∀r(k) = l ∈ T, are matrices that should be found. ∆V

g X

= x˜T (k + 1)(

π lm Pm )˜ x(k + 1) − x˜T (k)Pl x˜(k)

m=1

=

" w X

#T hi (s(k))A˜il x˜(k)

(

g X

" π lm Pm )

m=1

i=1

w X

# hj (s(k))A˜jl x˜(k)

j=1

T

−˜ x (k)Pl x˜(k) By using Lemma 2, it is obvious that " w # g X X ∆V ≤ x˜T (k) hi (s(k))A˜Til ( π lm Pm )A˜il x˜(k) − x˜T (k)Pl x˜(k) m=1

i=1

" T

= x˜ (k)

w X

hi (s(k)(A˜Til

= x˜T (k)

w X

# π lm Pm A˜il − Pl ) x˜(k)

m=1

i=1

"

g X

# hi (s(k)(A˜Til λl A˜il − Pl ) x˜(k)

(8)

i=1

Therefore, if (7) holds, by Schur complement, it is obvious that A˜Til λl A˜il − Pl < 0. Then we have ∆V < 0. According to the proof process shown in Theorem 1 of [34], we can conclude that lim E

N →∞

( N X

) xT (k, s(0))x(k, s(0))

<∞

k=1

Then the system is proved to be stochastically stable. This completes the proof. III. H∞

MODEL ERROR PERFORMANCE ANALYSIS

In this section, our first aim is to focus on the H∞ model error performance analysis for the model error system in (4). The following lemma presents an H∞ performance criterion for the underlying systems.

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Lemma 4: Consider the discrete-time fuzzy Markov systems with known transition probabilities in (4), assume that γ > 0 as a known constant. If there exist matrices Pl > 0, l ∈ T such that   ˜il −λl 0 λl A˜il λl B    ∗ −I C˜ ˜ il  D il   (9)  <0  ∗  ∗ −P 0 l   2 ∗ ∗ ∗ −γ I P where λl = m∈T π lm Pm , then the error system (4) is said to be stochastically stable with an H∞ performance index γ. Proof. If (9) holds, it follows from basic matrix theories that 7) holds. Therefore, the error system (4) is stochastically stable by Lemma 3. Now let (∞ ) X 4 J =E eT (k)e(k) − γ 2 uT (k)u(k) k=0

Under initial zero condition, it is obtained that V (x(k), r(k))|k=0 = 0, V (x(k), r(k))|k→∞ > 0. Further, we have J < J + V (∞) − V (0) (∞ ) X = E eT (k)e(k) − γ 2 uT (k)u(k) + 4V k=0

≤

 ∞ w X X k=0

where

 Γil = 



 x˜(k)   Γi  u(k)  

hi (s(k))

h

x˜T (k) uT (k)

i=1

C˜ilT C˜il + A˜Til λl A˜il − P l ∗

i

 T ˜ T ˜ ˜ ˜ Cil Dil + Ail λl Bil . ˜ T λl B ˜il ˜TD ˜ il −γ 2 I + B D il

By Schur complement, Γi < 0 is equivalent to  ˜ il C˜il D  −I   ∗ A˜Til λl A˜il − P l A˜Til λl  ˜ T λl B ˜il ∗ ∗ −γ 2 I + B il

il

   <0 

(10)

Similarly, inequality (10) is equal to (9). Therefore, if (9) holds, it is guaranteed that Γi < 0, which means J < 0. In this circumstance, (5) holds and the error system (4) is said to be stochastically stable with a H∞ performance index γ. This completes the proof.

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Remark 1: In fact, it is difficult to achieve reduced-order model by using Lemma 4, since the cross coupled terms that make unknown matrices hard to be solved are involved in different operation modes in (10). In order to overcome this difficulty, we give an improved version of lemma 4 by introducing slack matrix. Lemma 5: [35] Take system (4) into account and assume γ > 0 to be a known constant. If there exist Pl > 0, Rl , ∀l ∈ T, such that   ˜il λl − Rl − RlT 0 Rl A˜il Rl B     ˜ ˜ ∗ −I C D il il   (11)  <0  ∗ ∗ −Pl 0    ∗ ∗ ∗ −γ 2 I P where λl = m∈T π lm Pm . Then the system (4) is said to be stochastically stable with an H∞ performance index γ. Now, the following lemma is introduced for the derivation of our main result based on the projection approach. Lemma 6: [36] Assume W = W T ∈ Rt×t , U ∈ Rt×y and V ∈ Rz×t to be known matrices, and further assume that rank(U ) < t and rank(V ) < t. Consider the problem in terms of finding appropriate matrix η that satisfies W + U ηV + (U ηV )T < 0

(12)

then (12) is solvable for η if and only if U ⊥ W U ⊥T < 0,

V T ⊥ W V T ⊥T < 0

(13)

Moreover, if (13) holds, all the solutions of η can be given as η = UR+ ΨVL+ + Φ − UR+ UR ΦVL VL+ with

  −1 T T T −1 −1 1/2 T −1/2    Ψ = −Π UL ΛVR (VR ΛVR ) + Π Ξ L(VR ΛVR ) Λ = (UL Π−1 ULT − W )−1 > 0     Ξ = Π − U T (Λ − ΛV T (VR ΛV T )−1 VR Λ)UL > 0 L

R

R

where Φ, Π and L are matrices that have appropriate dimensions and satisfy Π > 0 and kLk < 0.

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IV. MAIN RESULTS In this section, two different methods are introduced to address the model approximation problem for FMJSs. The sufficient conditions on finding reduced-order models are presented based on convex linearization approach and projection approach, respectively. A: Convex Linearization Approach In this subsection, sufficient conditions will be established for finding admissible reduced order model with an H∞ performance index γ in the desired form of (3). Theorem 1: Consider the discrete-time fuzzy Markov system (2) and assume γ > 0 to be a given constant. If there exist appropriately dimensioned matrices P1l > 0, P3l > 0 and matrices ˇil , Cˇil , D ˇ il , ∀l ∈ T and i ∈ S, such that P2l , R1l , R2l , R3l , Aˇil , B   ˇil Λ11 Λ12 0 R1l Ail ΩAˇil R1l Bil + ΩB    ∗ Λ  ˇ ˇ 0 R A A R B + B 22 3l il il 3l il il      ∗  ˇ ˇ ∗ −I C − C D − D il il il il 4   Λ= (14) <0  ∗  ∗ ∗ −P1l −P2l 0      ∗  ∗ ∗ ∗ −P 0 3l   2 ∗ ∗ ∗ ∗ ∗ −γ I where

 4  T    Λ11 = λ1l − R1l − R1l 4 T Λ12 = λ2l − ΩR2l − R3l   4   Λ22 = λ3l − R2l − RT 2l

and Ω = [ I 0 ]T , I ∈ Rn×n , λ1l =

P

m∈T

π lm P1m , λ2l =

P

m∈T

π lm P2m , λ3l =

P

m∈T

π lm P3m ,

then we can find a reduced-order model and the model error system (4) is guaranteed to be stochastically stable with an H∞ performance index γ. Furthermore, if (14) is feasible, the system matrices can be given as follows:      −1 ˆ ˆ ˇ ˇ A Bil R 0 A Bil  il  =  2l   il  , l ∈ T, i ∈ S ˆ il ˇ il Cˆil D ∗ I Cˇil D

(15)

Proof. Take model error system (4) into account and assume the marices Pl , Rl to have following forms:

 Pl = 

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P1l P2l ∗

P3l





 , Rl = 

R1l ΩR2l R3l

R2l

 

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12

 Then one has λl = 

λ1l λ2l ∗

λ3l

 =

 P

m∈T

π lm 

P1m P2m ∗

P3m

 .

˜il , C˜il , D ˜ il , λl , Rl , Pl with the given augmented forms, and defining Then, by replacing A˜il , B ˇil = R2 B ˆil , Cˇil = Cˆil , D ˇ il = D ˆ il that bring convenience to computational Aˇil = R2 Aˆil , B process, we can obtain (14). In other words, if (14) holds, the model error system is said to be stochastically stable with an H∞ performance index γ. Moreover, if the inequality (14) is solvable, the parameters of the reduced order model can be set as (15). This completes the proof.

B. Projection approach Based on both Lemma 4 and Lemma 6, we aim to utilize another methodology to address the model reduction problem for the discrete-time fuzzy Markov system. Theorem 2: Consider the discrete-time model error system (4). The reduced order model in form of (3) such that the aforementioned model error system (4) is guaranteed to be stochastically stable with a prescribed H∞ performance index γ, if there exist matrices Pl > 0, l ∈ T such that



 T ¯ ¯ −H$ H H A H B l il il      ∗ −Pl 0 <0   2 ∗ ∗ −γ I   T ¯ −λ 0 λ A H l l il     T ¯ <0  ∗ −I Cil H   T ∗ ∗ −HPl H λl $ l = I

where λl =

P

m∈T

(16)

(17)

(18)

π lm Pm .

Furthermore, if the aforementioned inequalities are solvable, then the system matrices of the desired reduced-order model can be given as 

 ˆ ˆ Dil Cil  η il =  ˆil Aˆil B

with

February 6, 2015

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13 + + η il = UlR Ψil VL+ + Φil − UlR UlR Φil VL VL+

where  −1 T −1 1/2  T T −1 T −1/2    Ψil = −Πil UlL Λil VR (VR Λil VR ) + Πil Ξ Lil (VR Λil VR ) −1 T Λil = (UlL Π−1 >0 il UlL − Wil )     Ξil = Πil − U T (Λil − Λil V T (VR Λil V T )−1 VR Λil )UL > 0 iL

R

R

In the aforementioned equations, Φil , Πil and Lil are matrices that have appropriate dimensions and satisfy Πil > 0, kLil k < 0, and 

   ¯il  λl B       ¯ il    D       0        2  −γ I           

−λl 0 λl A¯il   −I C¯il 4  ∗ Wil =   ∗ ∗ −Pl  ∗ ∗ ∗  Ail 0n×t 4 A¯il =  0t×n 0t×t   B 4 ¯il =  il  B 0t×q h i 4 ¯ Cil = Cil 0p×t

4 ¯ il = D Dil h i 4 H = In×n 0n×t   0 0 n×p n×t 4  M1 =  0 It×t  t×p  0q×n 0q×t 4  N1 =  0t×n It×t h i 4 M2 = −Ip×p 0p×t   I q×q 4  N2 =  0t×q

(19)

                                                  

(20)

            h iT    4   Ul = M1T λTl M2T 0(p+t)×(n+t) 0(p+t)×q   h i   4  V = 0(q+t)×(n+t) 0(q+t)×p N1 N2

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˜il , C˜il and D ˜ il can be rewritten Proof. Using the matrices that are defined in Theorem 2, A˜il , B as follows:

4 A˜il = A¯il + M 1 ηN1 4 ˜il = ¯il + M 1 ηN2 B B 4   C˜il = C¯il + M 2 ηN1     ˜ 4 ¯ Dil = Dil + M 2 ηN2

      

Now, we are rewrite (9) as Wil + Ul η il V + (Ul η il V )T < 0

(21)

The definition of W , Ul and V in (21) can be found in (19) and (20). Since the orthogonal complement matrices of Ul and V are usually non-unique, we aim to find appropriate Ul⊥ and V ⊥ that are less complex for simulation and computation. In this    −1 0 0 0   Hλl  I 0    T⊥ Ul⊥ =  0 = 0 I 0 I 0 , V    0 0 0 0 0 I

case, we define  0 0   0 0   H 0

As indicated in Lemma 6, (21) is solvable for η if and only if Ul⊥ Wil Ul⊥T < 0,

V T ⊥ Wil V T ⊥T < 0

which can be specifically written as   −1 T ¯il H B ¯il −Hλ H H A l      ∗ −Pl 0 <0   2 ∗ ∗ −γ I   T ¯ −λ 0 λ A H l l il     T ¯  ∗ −I <0 Cil H   T ∗ ∗ −HPl H

(22)

(23)

(24)

As can be seen from (18) that λ−1 = $l . Therefore, it is obvious that (23) and (24) are equal l to (16) and (17), respectively. In addition, if (22)-(24) hold, an admissible solution to the system matrices of the reduced-order model can be achieved by utilizing the equalities given in Lemma 6. This makes the proof completed. In Theorem 2, we have derived sufficient conditions for the existence of H∞ model approximation for the studied system. It should be noticed that the obtained criterion is presented in February 6, 2015

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terms of LMI but with an equality constraint λl $l = I. Such problem can be solved by the so-called min-max algorithm, XY-centering algorithm and CCL algorithm [37]. Other effective algorithms that are established to address the non-convex problem can also be found in [8]. Compared to other potential methods or algorithms, CCL algorithm is proved to be an optimal choice since it is simple, efficient and reliable in numerical implementation. Therefore, we shall transform the non-convex feasibility problem that is shown in (18) into an optimization problem using CCL algorithm. It can be easily seen from Theorem 2 that the solution in terms of H∞ reduced-order model for discrete-time fuzzy markov systems is admissible if we can find P = (P1 , P2 , ..., Pg ), and $ = ($1 , $2 , ..., $g ), such that $l

X m∈T

π lm Pm = I

(25)

for all l ∈ T. CCL algorithm is proposed considering the fact that for any matrices ω > 0, P > 0, if the following LMI (26) holds, then trace(ωP ) ≥ n, where n is the dimension of matrix P . Moreover, ωP = I if and only if trace(ωP ) = n [18].  

ω I I P

 ≥0

(26)

Therefore, a feasible solution of (25) can be achieved by resolving the following minimization problem: Minimize

P

l∈T trace($ l

P

m∈T

π lm Pm )

subject to (16) and (17), and inequality   $l I   ≥ 0, ∀l ∈ T. P I π P m∈T lm m If and only if the optimal solution to inequality (27) satisfies trace($l

(27) P

m∈T

π lm Pm ) = n + t,

a solution to (25) is admissible. In this circumstance, the proposed H∞ model approximation problem has been converted to a minimization problem [8], [18]. CCL algorithm for H∞ model approximation for FMJSs Step 1: Find a feasible set ($ 0 , P0 ), where $ 0 = ($10 , $20 , ..., $g0 ), P0 = (P10 , P20 , ..., Pg0 ). It should be noted that the set of ($ 0 , P0 ) must satisfy (16), (17) and (27).

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Step 2: Assume $ k = ($1k , $2k , ..., $gk ) and Pk = (P1k , P2k , ..., Pgk ), set k = 0, then define 4

fk ($, P ) =

g g X X

π lm trace($lk Pm + $l Pmk )

l∈T m∈T

subject to (16), (17) and (27). Denote f ∗ as the optimal value of fk ($, P ). Step 3: If f ∗ converges to 2(n + t), specifically if (22) holds with |f ∗ − 2(n + t)| < δ for any sufficiently small scalar δ > 0, the obtained result is said to satisfy (25), then exit. Step 4: If k > K, which is the maximal iteration steps, exit. Then we say that we fail to find a reduced order for the original system. otherwise, set k = k + 1 and go to step 2. V. NUMERICAL EXAMPLE In this section, a numerical example is used to verify the effectiveness feasibility of the proposed model approximation approaches given in Theorem 1 and Theorem 2. Example 1: Consider a discrete-time T–S fuzzy Markov model error system with the form of (4) that involves two fuzzy membership functions and two separate modes. The system matrices of the original system are given as follows:    −2.8 −2.5 −0.4 −0.1 −3.1 −2.6 −0.4 −0.2     2.0  1.8 0.3 −0.15 0.2  0.2 −0.2 0.2    A11 =   , A12 =   −1.0 0.2 −0.1 0.2   −1.9 0.8 −0.15 0.4    0.1 1.0 0.3 −0.2 0.6 1.6 0.2 −0.4    −2.3 −3.0 −0.35 −0.2 −2.8 −2.5 −0.4 −0.1     1.4  2.0 0.1 −0.2 0.15  0.3 −0.15 0.2    A21 =   , A22 =   −2.8 0.7 −0.15 0.6   −1.0 0.2 −0.1 0.2    0.6 2.0 0.4 −1.0 0.1 1.0 0.3 −0.2         2.0 1.5 1.6 1.6          0.2   0.3   0.3   0.8          B11 =   , B12 =   , B21 =   , B22 =    0.2   −0.4   −1.4   0.3          0.6 0.3 0.6 0.3 h i h i C11 = −0.4 2.3 0.8 −0.3 , C12 = −0.3 2.2 0.9 −0.4 February 6, 2015

             

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17

C21 =

h

−0.7 1.5 0.5 0.5

i

, C22 =

h

−0.5 1.6 0.7 0.4

i

D11 = 0.5, D12 = 0.4, D21 = −0.4, D22 = −0.5 Regarding the Markov jump process, the transition rates between different operation models are assumed to be:

 Π=

0.6

0.4

0.45 0.55

 

As mentioned before, our main purpose is to find a reduced-order system with the form of (3) to precisely approximate the original system in (4). Specifically, the model error system to be found is stable and have guaranteed small system error with an optimal H∞ performance index compared with the original system. By employing the obtained techniques shown in aforementioned theorems, it is not difficult to obtain system matrices for the reduce-order system. Since the analysis process and simulation results derived by projection approach are similar with the ones shown above, the numerical results and relevant figures are omitted for the sake of brevity of the paper. In this example, given a desired reduced-order t = 2, the analysis result, which is presented in the following part, is obtained by solving (14) in Theorem 1. Moreover, the optimal H∞ performance index calculated by the convex linearization approach is γ min = 1.3358, which is relatively less than the attenuation level γ p min = 4.7102 derived by projection approach. In addition to this, it should be noticed that the computation for the projection approach is more complex than the first one. Thus, it is more likely to take more calculation time to achieve the system matrices if the projection approach is adopted. Therefore, it can be summarized that the convex linearization method is more effective and efficient than the technique based on projection approach. The system matrices for the reduced-order model derived by convex linearization approach is given as follows:    −3.4722 1.5433 −2.0719  , Aˆ12 =  Aˆ11 =  −0.3926 −1.8291 −0.5337    −2.7504 0.4047 −3.2329  , Aˆ22 =  Aˆ21 =  −0.1218 −1.6380 2.7141    1.4006 1.2148 ˆ11 =  ˆ12 =  ,B B −3.9615 −3.3251 February 6, 2015

0.2019 −1.9084 0.1547 −2.3660 

   



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18

 ˆ21 =  B Cˆ11 =

h

Cˆ21 =

h

0.9033 −2.3887





ˆ22 =  ,B

1.7556 −2.4906

i

2.1135 −2.0779

i

, Cˆ12 =

h

, Cˆ22 =

h

−1.4090 −2.1656

 

1.6784 −1.4852

i

0.3336 −1.9100

i

ˆ 11 = −0.5092, D ˆ 12 = −0.4133, D ˆ 21 = −0.5754, D ˆ 22 = −0.4008 D In order to verify the effectiveness and the feasibility of the proposed model approximation techniques, we will conduct performance analysis in terms of approximating ability of the reducedh iT order model. Assume initial state condition of the system to be zero, i.e. x(0) = 0 0 0 0 , h iT xˆ(0) = 0 0 . and let the membership function have the following forms:   h (x (k)) = 0.6 + sin(x (k)) 1 1 1  h2 (x1 (k)) = 0.4 − sin(x1 (k)) the disturbance input is with the following form u(k) = 0.5e−0.1k sin(0.5k), Figure 1 shows the output trajectories of the original system (depicted by dash-dot line) and the reduced-order system (depicted by red solid line), respectively, while Figure 2 describes the corresponding output error between the original system and the reduced-order system. Obviously, it can be said that for the aforementioned energy bounded disturbance u(k), the designed reduced-order system is stable from the output curved line in Figure 1. Therefore, it is concluded that the obtained system with reduced order is valid and feasible in terms of precisely √ approximating the Pk max

T

e (k)e(k) is original system. Moreover, it can be observed from Figure 3 that the ratio √Pkk=1 max T k=1

u (k)u(k)

constantly less than the attenuation level γ min = 1.3358, which also validates the effectiveness of the reduced-order model. VI. CONCLUSION In this paper, we have investigated the H∞ model approximation problem for a class of discrete-time T–S FMJSs. Two different approaches are employed to resolve the model approximation problem. Sufficient conditions in terms of a set of LMIs are derived for stability analysis with a guaranteed noise attenuation performance. To overcome the difficulty with respect to non-convex issues, CCL techniques are introduced to carry out the corresponding optimization task. Finally, testification results are given to illustrate the effectiveness and feasibility of the proposed methods. February 6, 2015

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1.2

output of the reduced−order system output of the original system

1 0.8

Outputs

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0

5

10

15

20 25 30 Time in samples

35

40

45

50

35

40

45

50

Fig. 1. Outputs of the original system and the reduced-order system

0.3

Output error between the two systems

0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 0

5

10

15

20 25 30 Time in samples

Fig. 2. Output error between the original system and the reduced-order system

ACKNOWLEDGEMENT The work was supported in part by National Natural Science Foundation of China (61021002, 61322301), State Key Laboratory of Robotics and System (HIT), Open Project of State Key Laboratory of Urban Water Resource and EnvironmentHarbin Institute of Technology (No. HCK201406), the Fundamental Research Funds for the Central Universities, China HIT.BRETIII.

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Ratio of error energy to disturbance energy

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

20 25 30 Time in samples

35

40

45

50

Fig. 3. The ratio of error energy to disturbance input energy of the system with reduced-order model

201211, HIT.BRETIV.201306. Also, this project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. (81-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU. R EFERENCES [1] W. Yan and J. Lam, “An approximate approach to H2 optimal model reduction,” IEEE Trans. Automat. Control, vol. 44, no. 7, pp. 1341–1358, 1999. [2] L. Zhang, P. Shi, E. K. Boukas, and C. Wang, “H∞ model reduction for uncertain switched linear discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2944–2949, 2008. [3] G. J. Lastman, “Reduced-order aggregated models for bilinear time-invariant dynamical systems,” IEEE Trans. Automat. Control, vol. 29, pp. 359–361, 1984. [4] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L∞ -error bounds,” Int. J. Control, vol. 39, no. 6, pp. 1115–1193, 1984. [5] B. Moore, “Principal component analysis in linear systems: controllability, observability, and model reduction,” IEEE Trans. Automat. Control, vol. 26, no. 5, pp. 17–31, 1981. [6] H. Yang, B. Jiang, and V. Cocquempot, “Fault tolerance analysis for stochastic systems using switching diffusion processes,” Int. J. Control, vol. 82, no. 8, pp. 1516–1525, 2009. [7] L. Zhang, E. K. Boukas, and P. Shi, “H∞ model reduction for discrete-time Markov jump linear systems with partially known transition probabilities,” Int. J. Control, vol. 82, no. 2, pp. 343–351, 2009. [8] X. Su, L. Wu, P. Shi, and Y. Song, “H∞ model reduction of Takagi-Sugeno fuzzy stochastic systems,” IEEE Trans. Systems, Man and Cybernetics - Part B, vol. 42, no. 6, pp. 1574–1585, 2012.

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